Theorem rnun 5152

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	|- ran ( A u. B ) = ( ran A u. ran B )

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 5149	. . . 4 |- `' ( A u. B ) = ( `' A u. `' B )
2	1	dmeqi 4938	. . 3 |- dom `' ( A u. B ) = dom ( `' A u. `' B )
3		dmun 4944	. . 3 |- dom ( `' A u. `' B ) = ( dom `' A u. dom `' B )
4	2, 3	eqtri 2252	. 2 |- dom `' ( A u. B ) = ( dom `' A u. dom `' B )
5		df-rn 4742	. 2 |- ran ( A u. B ) = dom `' ( A u. B )
6		df-rn 4742	. . 3 |- ran A = dom `' A
7		df-rn 4742	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 3361	. 2 |- ( ran A u. ran B ) = ( dom `' A u. dom `' B )
9	4, 5, 8	3eqtr4i 2262	1 |- ran ( A u. B ) = ( ran A u. ran B )

Syntax hints:   = wceq 1398   u. cun 3199   `'ccnv 4730   dom cdm 4731   ran crn 4732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by:  imaundi  5156  imaundir  5157  rnpropg  5223  fun  5516  foun  5611  fpr  5844  fprg  5845  sbthlemi6  7204  exmidfodomrlemim  7455